665 research outputs found
Fractal Spectrum of a Quasi_periodically Driven Spin System
We numerically perform a spectral analysis of a quasi-periodically driven
spin 1/2 system, the spectrum of which is Singular Continuous. We compute
fractal dimensions of spectral measures and discuss their connections with the
time behaviour of various dynamical quantities, such as the moments of the
distribution of the wave packet. Our data suggest a close similarity between
the information dimension of the spectrum and the exponent ruling the algebraic
growth of the 'entropic width' of wavepackets.Comment: 17 pages, RevTex, 5 figs. available on request from
[email protected]
Analysis of Daily Streamflow Complexity by Kolmogorov Measures and Lyapunov Exponent
Analysis of daily streamflow variability in space and time is important for
water resources planning, development, and management. The natural variability
of streamflow is being complicated by anthropogenic influences and climate
change, which may introduce additional complexity into the phenomenological
records. To address this question for daily discharge data recorded during the
period 1989-2016 at twelve gauging stations on Brazos River in Texas (USA), we
use a set of novel quantitative tools: Kolmogorov complexity (KC) with its
derivative associated measures to assess complexity, and Lyapunov time (LT) to
assess predictability. We find that all daily discharge series exhibit long
memory with an increasing downflow tendency, while the randomness of the series
at individual sites cannot be definitively concluded. All Kolmogorov complexity
measures have relatively small values with the exception of the USGS (United
States Geological Survey) 08088610 station at Graford, Texas, which exhibits
the highest values of these complexity measures. This finding may be attributed
to the elevated effect of human activities at Graford, and proportionally
lesser effect at other stations. In addition, complexity tends to decrease
downflow, meaning that larger catchments are generally less influenced by
anthropogenic activity. The correction on randomness of Lyapunov time
(quantifying predictability) is found to be inversely proportional to the
Kolmogorov complexity, which strengthens our conclusion regarding the effect of
anthropogenic activities, considering that KC and LT are distinct measures,
based on rather different techniques
Divergent Predictive States: The Statistical Complexity Dimension of Stationary, Ergodic Hidden Markov Processes
Even simply-defined, finite-state generators produce stochastic processes
that require tracking an uncountable infinity of probabilistic features for
optimal prediction. For processes generated by hidden Markov chains the
consequences are dramatic. Their predictive models are generically
infinite-state. And, until recently, one could determine neither their
intrinsic randomness nor structural complexity. The prequel, though, introduced
methods to accurately calculate the Shannon entropy rate (randomness) and to
constructively determine their minimal (though, infinite) set of predictive
features. Leveraging this, we address the complementary challenge of
determining how structured hidden Markov processes are by calculating their
statistical complexity dimension -- the information dimension of the minimal
set of predictive features. This tracks the divergence rate of the minimal
memory resources required to optimally predict a broad class of truly complex
processes.Comment: 16 pages, 6 figures; Supplementary Material, 6 pages, 2 figures;
http://csc.ucdavis.edu/~cmg/compmech/pubs/icfshmp.ht
Unified approach to catastrophic events: from the normal state to geological or biological shock in terms of spectral fractal and nonlinear analysis
An important question in geophysics is whether earthquakes (EQs) can be anticipated prior to their occurrence. Pre-seismic electromagnetic (EM) emissions provide a promising window through which the dynamics of EQ preparation can be investigated. However, the existence of precursory features in pre-seismic EM emissions is still debatable: in principle, it is difficult to prove associations between events separated in time, such as EQs and their EM precursors. The scope of this paper is the investigation of the pre-seismic EM activity in terms of complexity. A basic reason for our interest in complexity is the striking similarity in behavior close to irreversible phase transitions among systems that are otherwise quite different in nature. Interestingly, theoretical studies (Hopfield, 1994; Herz and Hopfield 1995; Rundle et al., 1995; Corral et al., 1997) suggest that the EQ dynamics at the final stage and neural seizure dynamics should have many similar features and can be analyzed within similar mathematical frameworks. Motivated by this hypothesis, we evaluate the capability of linear and non-linear techniques to extract common features from brain electrical activities and pre-seismic EM emissions predictive of epileptic seizures and EQs respectively. The results suggest that a unified theory may exist for the ways in which firing neurons and opening cracks organize themselves to produce a large crisis, while the preparation of an epileptic shock or a large EQ can be studied in terms of ''Intermittent Criticality''
An elementary way to rigorously estimate convergence to equilibrium and escape rates
We show an elementary method to have (finite time and asymptotic) computer
assisted explicit upper bounds on convergence to equilibrium (decay of
correlations) and escape rate for systems satisfying a Lasota Yorke inequality.
The bounds are deduced by the ones of suitable approximations of the system's
transfer operator. We also present some rigorous experiment showing the
approach and some concrete result.Comment: 14 pages, 6 figure
Multivariate multiscale complexity analysis
Established dynamical complexity analysis measures operate at a single scale and thus fail
to quantify inherent long-range correlations in real world data, a key feature of complex
systems. They are designed for scalar time series, however, multivariate observations are
common in modern real world scenarios and their simultaneous analysis is a prerequisite for
the understanding of the underlying signal generating model. To that end, this thesis first
introduces a notion of multivariate sample entropy and thus extends the current univariate
complexity analysis to the multivariate case. The proposed multivariate multiscale entropy
(MMSE) algorithm is shown to be capable of addressing the dynamical complexity of such
data directly in the domain where they reside, and at multiple temporal scales, thus
making full use of all the available information, both within and across the multiple data
channels. Next, the intrinsic multivariate scales of the input data are generated adaptively
via the multivariate empirical mode decomposition (MEMD) algorithm. This allows for
both generating comparable scales from multiple data channels, and for temporal scales
of same length as the length of input signal, thus, removing the critical limitation on
input data length in current complexity analysis methods. The resulting MEMD-enhanced
MMSE method is also shown to be suitable for non-stationary multivariate data analysis
owing to the data-driven nature of MEMD algorithm, as non-stationarity is the biggest
obstacle for meaningful complexity analysis. This thesis presents a quantum step forward
in this area, by introducing robust and physically meaningful complexity estimates of
real-world systems, which are typically multivariate, finite in duration, and of noisy and
heterogeneous natures. This also allows us to gain better understanding of the complexity
of the underlying multivariate model and more degrees of freedom and rigor in the analysis.
Simulations on both synthetic and real world multivariate data sets support the analysis
Generalization Bounds with Data-dependent Fractal Dimensions
Providing generalization guarantees for modern neural networks has been a
crucial task in statistical learning. Recently, several studies have attempted
to analyze the generalization error in such settings by using tools from
fractal geometry. While these works have successfully introduced new
mathematical tools to apprehend generalization, they heavily rely on a
Lipschitz continuity assumption, which in general does not hold for neural
networks and might make the bounds vacuous. In this work, we address this issue
and prove fractal geometry-based generalization bounds without requiring any
Lipschitz assumption. To achieve this goal, we build up on a classical covering
argument in learning theory and introduce a data-dependent fractal dimension.
Despite introducing a significant amount of technical complications, this new
notion lets us control the generalization error (over either fixed or random
hypothesis spaces) along with certain mutual information (MI) terms. To provide
a clearer interpretation to the newly introduced MI terms, as a next step, we
introduce a notion of "geometric stability" and link our bounds to the prior
art. Finally, we make a rigorous connection between the proposed data-dependent
dimension and topological data analysis tools, which then enables us to compute
the dimension in a numerically efficient way. We support our theory with
experiments conducted on various settings
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